Final answer:
In solving the polynomial f(x) = x³ + 10x² + 23x + 14, the possible rational zeros are ± 1, ± 2, ± 7, and ± 14. The actual rational zeros found are -1, -2, and -7. The final factored form of f(x) is (x + 1)(x + 2)(x + 7).
Step-by-step explanation:
The task given involves a polynomial function f(x) = x³ + 10x² + 23x + 14. We need to (a) list all possible rational zeros, (b) find all rational zeros, and (c) factor f(x).
(a) Listing all possible rational zeros:
Possible rational zeros of a polynomial can be found using the Rational Zero Theorem. The theorem states that any rational zero, which is of the form p/q (where p and q are integers and q is not 0), is a fraction where p is a factor of the constant term (14 in this case) and q is a factor of the leading coefficient (1 in this case). Therefore, the possible rational zeros are ± 1, ± 2, ± 7, and ± 14.
(b) Finding all rational zeros:
To find the actual rational zeros, we can use synthetic division or polynomial division and test each possible zero. After completing synthetic division for each possible zero, it turns out that -1, -2, and -7 are zeros of the polynomial.
(c) Factoring f(x):
Now that we have found the zeros, we can factor the polynomial. Given the zeros -1, -2, and -7, we can write the factors as (x + 1), (x + 2), and (x + 7). Thus, f(x) can be factored as (x + 1)(x + 2)(x + 7).